In statistics, local asymptotic normality is a property of a sequence of statistical models, which allows this sequence to be asymptotically approximated by a normal location model, after an appropriate rescaling of the parameter. An important example when the local asymptotic normality holds is in the case of i.i.d sampling from a regular parametric model.
The notion of local asymptotic normality was introduced by and is fundamental in the treatment of estimator and test efficiency.
A sequence of parametric statistical models } is said to be locally asymptotically normal (LAN) at ø if there exist matrices r<sub>n</sub> and I<sub>ø</sub> and a random vector such that, for every converging sequence ,
where the derivative here is a RadonâÂÂNikodym derivative, which is a formalised version of the likelihood ratio, and where o is a type of little O in probability notation. In other words, the local likelihood ratio must converge in distribution to a normal random variable whose mean is equal to minus one half the variance:
The sequences of distributions and are contiguous.
The most straightforward example of a LAN model is an iid model whose likelihood is twice continuously differentiable. Suppose } is an iid sample, where each X<sub>i</sub> has density function . The likelihood function of the model is equal to
If f is twice continuously differentiable in ø, then
Plugging in , gives
By the central limit theorem, the first term (in parentheses) converges in distribution to a normal random variable , whereas by the law of large numbers the expression in second parentheses converges in probability to I<sub>ø</sub>, which is the Fisher information matrix:
Thus, the definition of the local asymptotic normality is satisfied, and we have confirmed that the parametric model with iid observations and twice continuously differentiable likelihood has the LAN property.